Optimal routing and buffer allocation for a class of finite capacity queueing systems

The problem of routing jobs to K parallel queues with identical exponential servers and unequal finite buffer capacities is considered. Routing decisions are taken by a controller which has buffering space available to it and may delay routing of a customer to a queue. Using ideas from weak majorization, it is shown that the shorter nonfull queue delayed (SNQD) policy minimizes both the total number of customers in the system at any time and the number of customers that are rejected by that time. The SNQD policy always delays routing decisions as long as all servers are busy. Only when all the buffers at the controller are occupied is a customer routed to the queue with the shortest queue length that is not at capacity. Moreover, it is shown that, if a fixed number of buffers is to be distributed among the K queues, then the optimal allocation scheme is the one in which the difference between the maximum and minimum queue capacities is minimized, i.e., becomes either 0 or 1.